"D'oh! Negatives!"
Jul. 20th, 2004 07:28 pmSo I got a new ATM card. Replacements are free, apparently.
Today during lecture Dan suddenly threw up. I was sitting right in front of him, but he just threw up on the ground in front of him, so I didn't get hit.
More ideas for "Monty Python Does the Integers" have come up, most not very good. We are so doing the WOP one, though.
Apparently Evan will be chugging a bottle of corn syrup. I'm told he's done this before. Some people wonder, though, if he's actually the person who signed him up...
One act listed is "Impersonation of Dustin". The person doing it? "Dan".
Haha, here's a problem I messed up on. P14 on geometry pset 5 asks about the function F that maps 2x2 matrices over R to their corresponding fractional linear transformations. Is it a homomorphism? Yes, that's easy... Is it injective?
Now at first I was going to say "no", as I realized that F(kA)=F(A), but then I noticed that the thing actually defined F:SL(2,R)→L (L being the group of fractional linear transformations with "determinant" >0.). And multiplying by a constant would change the determinant, right? So I start writing up a proof that it *is* injective. And it's not till I get to the very end that I get to the hole in the proof: since these are 2x2 matrices, det(kA)=k²det(A), so if A∈SL(2,R), so is -A. And of course F(A)=F(-A). It's too much trouble to erase what I already had and it would look awful to write over anyway, so I just put a box around it and put a big X over it.
Is it surjective? Not too hard... is the image a subgroup of L - What do you think?! (Especially since it *is* surjective...)
Then it asks, what's the kernel of F? (Though it doesn't call it that, we're not expected to know the word "kernel", don't know why it doesn't just define it there. :P )
...and I realize that I and -I are the only 2 matrices in the kernel, and that if I want to show that, what I should do is use the stuff in the big box I just crossed out. So I keep the box but erase the big X, and put a label below it, "The temporarily-crossed-out box." And I write, "Since F(I)=Id, by the temporarily-crossed-out box, F(A)=Id ⇒ A=I or A=-I." Yes, I really did write "by the temporarily-crossed-out-box".
Today in Graphs and Knots Rohrlich introduced Conway polynomials, and mentioned that they were - what was the word, renormalizations? - of Alexander polynomials. After class I asked what the Alexander polynomial was. He didn't exactly recall, but I must say, that was one awful-sounding outline of a definition. You first take some group that is somehow associated with the knot (he didn't say just how), take a presentation of that group, formally differentiate the relations somehow (!), and then from this you do stuff that eventually gets you a matrix of polynomials... and I don't remember what you do with that - take the determinant? :-/ Still, that sounds *awful*.
There was actually something of a food fight at lunch today. Something of one. Essentially, Xiao threw Coke on ODan, and ODan threw ketchup on Xiao, and lots of other stuff got hit, and really, that was it. And all this happened while I was away from the table. :P
-Sniffnoy
--
"What I'm saying is Sluggy's going to be like a box o chocolates. You
never know what yer gunna get! I'm personally hoping for "chocolates,"
only because that's what it says on the box."
-Pete Abrams
Today during lecture Dan suddenly threw up. I was sitting right in front of him, but he just threw up on the ground in front of him, so I didn't get hit.
More ideas for "Monty Python Does the Integers" have come up, most not very good. We are so doing the WOP one, though.
Apparently Evan will be chugging a bottle of corn syrup. I'm told he's done this before. Some people wonder, though, if he's actually the person who signed him up...
One act listed is "Impersonation of Dustin". The person doing it? "Dan".
Haha, here's a problem I messed up on. P14 on geometry pset 5 asks about the function F that maps 2x2 matrices over R to their corresponding fractional linear transformations. Is it a homomorphism? Yes, that's easy... Is it injective?
Now at first I was going to say "no", as I realized that F(kA)=F(A), but then I noticed that the thing actually defined F:SL(2,R)→L (L being the group of fractional linear transformations with "determinant" >0.). And multiplying by a constant would change the determinant, right? So I start writing up a proof that it *is* injective. And it's not till I get to the very end that I get to the hole in the proof: since these are 2x2 matrices, det(kA)=k²det(A), so if A∈SL(2,R), so is -A. And of course F(A)=F(-A). It's too much trouble to erase what I already had and it would look awful to write over anyway, so I just put a box around it and put a big X over it.
Is it surjective? Not too hard... is the image a subgroup of L - What do you think?! (Especially since it *is* surjective...)
Then it asks, what's the kernel of F? (Though it doesn't call it that, we're not expected to know the word "kernel", don't know why it doesn't just define it there. :P )
...and I realize that I and -I are the only 2 matrices in the kernel, and that if I want to show that, what I should do is use the stuff in the big box I just crossed out. So I keep the box but erase the big X, and put a label below it, "The temporarily-crossed-out box." And I write, "Since F(I)=Id, by the temporarily-crossed-out box, F(A)=Id ⇒ A=I or A=-I." Yes, I really did write "by the temporarily-crossed-out-box".
Today in Graphs and Knots Rohrlich introduced Conway polynomials, and mentioned that they were - what was the word, renormalizations? - of Alexander polynomials. After class I asked what the Alexander polynomial was. He didn't exactly recall, but I must say, that was one awful-sounding outline of a definition. You first take some group that is somehow associated with the knot (he didn't say just how), take a presentation of that group, formally differentiate the relations somehow (!), and then from this you do stuff that eventually gets you a matrix of polynomials... and I don't remember what you do with that - take the determinant? :-/ Still, that sounds *awful*.
There was actually something of a food fight at lunch today. Something of one. Essentially, Xiao threw Coke on ODan, and ODan threw ketchup on Xiao, and lots of other stuff got hit, and really, that was it. And all this happened while I was away from the table. :P
-Sniffnoy
--
"What I'm saying is Sluggy's going to be like a box o chocolates. You
never know what yer gunna get! I'm personally hoping for "chocolates,"
only because that's what it says on the box."
-Pete Abrams
